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06_ts_chap4 - Outline Sample ACF ARMA ARIMA Residual...

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Outline Sample ACF ARMA & ARIMA Residual analysis R example Estimating Time-Series Models AMS316, Stony Brook University Outline Sample ACF ARMA & ARIMA Residual analysis R example Outline 1 Estimating ACVFs and ACFs 2 Estimating parameters of an ARMA and ARIMA model 3 Residual analysis 4 R implementation AMS316, Stony Brook University
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Outline Sample ACF ARMA & ARIMA Residual analysis R example Estimating ACVFs and ACFs Suppose we have N observations on a stationary process, say x 1 , x 2 , . . . , x N . Then the sample autocovariance at lag k is given by c k := γ ( k ) = 1 N N - k t =1 ( x t - ¯ x )( x t + k - ¯ x ) , is the usual estimator for the autocovariance γ ( k ) at lag k . An alternative estimator of γ ( k ) is c k = 1 N - k N - k t =1 ( x t - ¯ x )( x t + k - ¯ x ) , Having estimated the acv.f., we then take r k := ρ ( k ) = c k /c 0 as an estimator for ρ ( k ) . AMS316, Stony Brook University Outline Sample ACF ARMA & ARIMA Residual analysis R example Suppose that x 1 , . . . , x N are observations on independent and identically distributed random variables with arbitrary mean, then we have, (The proof is beyond our discussion) E ( r k ) ≈ - 1 /N, Var ( r k ) 1 /N and that r k is asymptotically normally distributed under weak conditions. We can check for randomness by plotting approximate 95% confidence limits at - 1 /N ± 2 / N , which are often further approximated to ± 2 / N . Observed values of r k which fall outside these limits are significantly di ff erent from 0 at the 5% level.
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